Design of Design of Regular Regular Quantum Quantum Circuits Circuits Regular circuit = tile- based circuit
Design of Regular Design of Regular Quantum CircuitsQuantum Circuits
Regular circuit = tile-based circuit
REVERSIBLE REVERSIBLE LOGICLOGIC
2
Reversible Permutative logic Gates and Circuits
A logic gate is reversible if Each input is mapped to a unique output It permutes the set of input values
A combinational logic circuit is reversible if it satisfies the following:
Has only one Fanout, Uses only reversible gates, No feedback path, has as many input wires as output wires, and permutes the
input values.
3
Basic Reversible Gates
a a
aa
bbac
4
NOT gate
a b a c0 0 0 00 1 0 11 0 1 11 1 1 0
Controlled-NOT or Feynman gate
Basic Reversible Gates
a
c
b
a
b
cabf
5
a b c a b f0 0 0 0 0 00 0 1 0 0 10 1 0 0 1 00 1 1 0 1 11 0 0 1 0 01 0 1 1 0 11 1 0 1 1 11 1 1 1 1 0
Toffoli gate (Controlled-Controlled NOT gate)
Basic Reversible Gates
a
a
b
b
6
Swap gate
Implementation of Swap gate using controlled-NOT
Basic Reversible Gates
a
b
c
baacf
abcag
a
b
c
baacf
abcag
7
a b c a f g0 0 0 0 0 00 0 1 0 0 10 1 0 0 1 00 1 1 0 1 11 0 0 1 0 01 0 1 1 1 01 1 0 1 0 11 1 1 1 1 1
Fredkin gate (Controlled SWAP gate)
ALGORITHMS FOR SYNTHESIS ALGORITHMS FOR SYNTHESIS OF REVERSIBLE LOGIC CIRCUITSOF REVERSIBLE LOGIC CIRCUITS
8
Popular Algorithms for Synthesis of Reversible Logic Circuits
MMD: Transformation based
Gupta-Agrawal-Jha: PPRM based
Mishchenko-Perkowski: Reversible wave cascade
Kerntopf: Heuristics based
Wille: BDD based synthesis
9
reed-mulLER reed-mulLER EXPANSION IN EXPANSION IN SYNTHESIS OF SYNTHESIS OF REVERSIBLE REVERSIBLE
CIRCUITSCIRCUITS10
IDEA: use reed-mulLER IDEA: use reed-mulLER EXPANSION IN SYNTHESIS OF EXPANSION IN SYNTHESIS OF
REVERSIBLE CIRCUITSREVERSIBLE CIRCUITSA New Representation is Reed-Muller Expansion
(Positive Polarity Reed-Muller).
This idea appeared for the first time in paper of Aggrawal and Jha, this paper was a competitor to MMD algorithm.
Now we design a new algorithm which takes into account multi-level expansion for reversible circuits.
11
Example of Agrawal-Jha Algorithm
12
c b a co bo ao
0 0 0 0 0 1
0 0 1 0 0 0
0 1 0 1 1 1
0 1 1 0 1 0
1 0 0 0 0 1
1 0 1 1 0 0
1 1 0 1 0 1
1 1 1 1 1 0
PPRM form for each output in terms ofInput variables are given as follows and node is created
PPRM form for each output in terms ofInput variables are given as follows and node is created
acabbc 0
accbb 0
10 aa
•Reversible function specification is given as a truth table shown here•Output c0, b0 and a0 are derived using EXORCISM-2 developed at PSU and parent node is created
Agrawal-Jha Algorithm (cont..)
Parent node is explored by examining each output variable in the PPRM expansion.
Factors are searched in the PPRM expansions that do not contain the same input variable.
For example in the expansion below appropriate terms are “c” and “ac”
The substitution is performed as In this example OR
accbb 0
factorvv ii
13
cbb acbb
Agrawal-Jha Algorithm (cont..)
acabbc 0
accbb 0
10 aa
Node0
PQ head Node0
1aa
Algorithm identifies three possible substitutions
1. 2. 3.
cbb acbb
14
Agrawal-Jha Algorithm (cont..)
1aa acbb cbb
aa 0
acbb 0
acabbc 0
10 aa
accbb 0
acabbc 0
acabbc 0
cbb 0
10 aa
abbcc 0
acbb 0
10 aa
PQ head Node 1.0 Node 1.1 Node 1.2
Node 1.0 Node 1.1 Node 1.2
15
New nodes are created based on substitution
16
1aa cbb
aa 0
acbb 0
acabbc 0
10 aa
accbb 0
acabbc 0
acabbc 0
cbb 0
10 aa
abbcc 0
acbb 0
10 aa
Node 1.0
Node 1.1 Node 1.2
acbb
acbb abcc
abcc 0
bb 0
aa 0
acabcc 0
acabbb 0
aa 0
Node 2.0 Node 2.1
PQ head Node 2.0 Node 1.1 Node 1.2 Node 2.1
Next stage of Aggrawal-Jha algorithm
17
1aa cbb
aa 0
acbb 0
acabbc 0
10 aa
accbb 0
acabbc 0
acabbc 0
cbb 0
10 aa
abbcc 0
acbb 0
10 aa
Node 1.0
Node 1.1 Node 1.2
acbb
acbb abcc
abcc 0
bb 0
aa 0
acabcc 0
acabbb 0
aa 0
Node 2.0 Node 2.1
abcc
00 cc bb 0
aa 0
Node 3.0
PQ head Node 1.1 Node 1.2 Node 2.1
Next stage of Aggrawal-Jha algorithm
18
1aa
aa 0
acbb 0
acabbc 0
10 aa
accbb 0
acabbc 0
Node 1.0
acbb
abcc 0
bb 0
aa 0
Node 2.0
abcc
00 cc bb 0
aa 0
Node 3.0
Node0
a
c
b acbb
a
b
c abcc
a
b
c
ao
bo
co
Final circuit
Solution found by the Aggrawal-Jha algorithm
Problem with Current Synthesis Approaches
Common problem with current approaches: they invariably use nxn Toffoli gates, that might imposes technological limitations.
High Quantum cost of Toffoli gates with many inputs.
Synthesize only reversible functions, not Boolean functions that is not reversible.
19
Quantum Cost of 4x4 Toffoli Gate
20
Implementation of 4x4 Toffoli gate with Quantum realizable 2x2 primitives such as controlled-V, controlled-NOT, controlled-V+.
Implementation of 4x4 Toffoli gate with Quantum realizable 2x2 primitives such as controlled-V, controlled-NOT, controlled-V+.
a
b
0
c
d
a
b
d
c
V V V+
V V V+
V V V+
a
b
0
c
d
CREATING CREATING QUANTUM QUANTUM
ARRAY FROM ARRAY FROM LATTICELATTICE
21
Expansions Rules for Lattice DIAGRAAMS
22
zbby
r sbb b b
x y z v
r sbb b b
xv
RULE (S, S)
r sb1 b
x y z v
r sb1 b
x
zbby
RULE (pD, pD)1
vzy
1
r sb1
x y z v
r sb b
x
zbby
RULE (nD, nD)1
vzy
b 11
r sb
x y z v
r sb
x
zbby
RULE (s, pD)1
vzy
1b b b b
r sb
x y z v
r sb
x
zbby
RULE (pD, s)1
vzy
1b b b b
Positive Davio Tree can be created by expanding PPRM function using positive Davio expansion.
Positive Davio Lattice is created by performing joining operation for neighboring cells at every level.
Other Lattices can be created using similar method but using expansions such as Shannon or Negative Davio expansions or combination of them.
Creating Quantum Array from Lattices
On the previous foils I showed representation of the Davio and Shannon cells as cascade of reversible gates.
Next I present unique method to create Quantum Array from Positive Davio Lattice.
The same approach can be used for other Lattices.
23
Creating Positive Davio Lattice
a
c
b
a
b
cabd
a
bc cabd
Positive Davio cell
Positive Davio cell representation with
Toffoli gate
24
bcdcdbcacabdbdad 1
abdbdad 1 bddba
abdba abb 1
ad1 a1
d
a
1 c
1 d 1 d
1 b 1 b
1 a 1 a 1 a
1 d
1
11
1
1 1
10
0
Each node represents pDv cell.
Creating Quantum Array from Positive Davio Lattice
25
+
+ +
++
+ ++
+
0
d
1
11
1
1
1
1 1
1
111
1
d
a
d
bb
1
a a
c
1
Quantum Array Representation
26
abcd
0
1
1
1
0
1
a1
ad1
bab1
d
a aabdb
adabddb1
bddab
bcdcdacbcabdaddb1
garbage
garbage
garbage
garbage
garbage
function
Quantum Array Representation
27
abcd
0
1
1
1
0
1
a1
ad1
bab1
d
a aabdb
adabddb1
bddab
bcdcdacbcabdaddb1
garbage
garbage
garbage
garbage
garbage
function
Creating Positive Davio Lattice
a
c
b
a
b
cabd
a
bc cabd
Positive Davio cell
Positive Davio cell representation with
Toffoli gate
28
Each node represents pDv cell.
Quantum Array Representation
29
Advantages of Lattice to QA
Reversible circuit synthesized with only 3x3 Toffoli gates.
Generates reversible circuit for any ESOP.
Adds ancilla bits but overall cost of the circuit will be lower due to use of low cost 3x3 Toffoli gates.
30
31
Calculating Single-Output Shannon Lattice for Calculating Single-Output Shannon Lattice for Completely Specified Boolean Function.Completely Specified Boolean Function.
32
Calculating Multi-Calculating Multi-Output Shannon Output Shannon
Lattice for Lattice for Completely Completely Specified Specified Boolean Boolean
Function.Function.
33
Calculating Multi-Output Shannon Calculating Multi-Output Shannon Lattice for Completely Specified Boolean Lattice for Completely Specified Boolean
Function.Function.
DIPAL GATES, DIPAL GATES, DIPAL GATE DIPAL GATE
FAMILIES AND FAMILIES AND THEIR ARRAYSTHEIR ARRAYS
34
Representation of pdv cell as a toffoli gate
a
c
b
a
b
cabd
a
b
c cabd
a
c
b
a
ba
b
c
cbad cbad
Positive Davio cell
Positive Davio cell representation with Toffoli
gate
Negative Davio cell
Negative Davio cell representation with Toffoli
gate
35
Development of Dipal gate
][
]1[
cbab
acabb
acba
acbaf
36
a
b
c
cabaf a
b
c
cabaf
cb
a
Shannon cell
Dipal cell representation with
reversible gates
There are 23! = 8! = 40320 3x3 Reversible logic functions, however only handful of them shown earlier are useful for synthesis purpose.
Dipal gate is a reversibleequivalent of Shannon cell Dipal gate is a reversibleequivalent of Shannon cell
•Find the reversible counterpart of well-known structures BDD, Lattices, KFDD•Show Dipal cell is between Toffoli and Fredkin
Development of Dipal gate (cont..)
37
a
b
c
bacaf
a
b
c
a
cb
bacaf
Shannon cell with negative variable
Dipal cell with negative variable represented with
reversible gates
Development of Dipal gate
][
]1[
cbab
acabb
acba
acbaf
38
a
b
c
cabaf a
b
c
cabaf
cb
a
Shannon cell
Dipal cell representation with
reversible gates
There are 23! = 8! = 40320 3x3 Reversible logic functions, however only handful of them shown earlier are useful for synthesis purpose.
Dipal gate is a reversibleequivalent of Shannon cell
Dipal gate is a reversibleequivalent of Shannon cell
Dipal gate truth table
c b a a
0 0 0 0 0 0
0 0 1 0 0 1
0 1 0 1 1 0
0 1 1 1 0 1
1 0 0 1 0 0
1 0 1 1 1 1
1 1 0 0 1 0
1 1 1 0 1 1
39
b c
b a[b c] input
output
0 0
1 1
2 6
3 5
4 4
5 7
6 2
7 3
Dipal gate unitary matrix
40
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 1 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
000
001
010
011
100
101
110
111
000 001 010 011 100 101 110 111
Variants of Dipal gates
41
This is called a Dipal Gate Family
General view of Dipal Family Gate
EXPERIMENTAL EXPERIMENTAL RESULTSRESULTS
42
Results with Pdv Lattice and comparison with MMD and AJ results
43
Benchmark #Real inputs
#Garbage inputs
#Gates Lattice
Cost Lattice
CPU time Lattice
#Gates DMM
Cost DMM
#Gates AJ
Cost AJ
2to5 5 4 31 107 0.12 15 107 20 100
rd32 3 1 4 8 < 0.01 4 8 4 8
rd53 5 5 11 39 < 0.01 16 75 13 116
3_17 3 1 10 21 < 0.01 6 12 6 14
6sym 10 6 34 150 0.37 20 62 NA NA
5mod5 5 1 14 58 < 0.01 10 90 11 91
4mod5 4 1 6 18 < 0.01 5 13 5 13
ham3 3 0 3 7 < 0.01 5 7 5 9
xor5 5 0 4 4 < 0.01 4 4 4 4
Xnor5 5 1 5 5 < 0.01 -------- ---------- ---------- ----------
decod24 4 2 10 30 < 0.01 -------- ---------- 11 31
Cycle10_2 12 6 180 860 27.9 19 1198 ---------- ----------
ham7 7 5 22 58 0.10 23 81 24 68
Results with Pdv Lattice and comparison with MMD and AJ results (cont..)
44
Benchmark #Real inputs
#Garbage inputs
#Gates Lattice
Cost Lattice
CPU time Lattice
#Gates DMM
Cost DMM
#Gates AJ
Cost AJ
graycode6 6 5 5 5 < 0.01 5 5 5 5
graycode10 10 9 9 9 < 0.01 9 9 9 9
graycode20 20 19 19 19 < 0.01 19 19 19 19
nth_prime3_inc
3 4 4 6 < 0.01 4 6 ---------- ----------
nth_prime4_inc
4 5 16 48 < 0.01 12 58 ---------- ----------
nth_prime5_inc
5 5 29 91 0.22 26 78 ---------- ----------
alu 5 2 5 17 < 0.01 -------- ---------- 18 114
4_49 4 4 16 52 0.04 16 58 13 61
hwb4 4 4 12 28 < 0.01 17 63 15 35
hwb5 5 5 24 96 1.2 24 104 ---------- ----------
hwb6 6 6 32 128 2.0 42 140 ---------- ----------
pprm1 4 4 9 33 < 0.01 -------- ---------- ---------- ----------
Results with shannon Lattice
45
Benchmark #Inputs #Gates pDv Lattice
Cost pDv Lattice
#Gates Shannon Lattice
Cost Shannon Lattice
2to5 5 31 107 41 117
rd32 3 4 8 4 8
rd53 5 11 39 18 46
3_17 3 10 21 15 26
6sym 10 34 150 51 167
5mod5 5 14 58 30 81
4mod5 4 6 18 12 24
Ham3 3 3 7 6 10
xor5 5 4 4 4 4
Xnor5 5 5 5 5 5
Decod24 4 10 30 20 40
Cycle10_2 12 180 860 270 950
Ham7 7 22 58 32 68
Results with shannon Lattice (cont..)
46
Benchmark #Inputs #Gates pDv Lattice
Cost pDv Lattice
#Gates Shannon Lattice
Cost Shannon Lattice
Graycode6 6 5 5 5 5
Graycode10 10 9 9 9 9
Graycode20 20 19 19 19 19
nth_prime3_inc
3 4 6 6 8
nth_prime4_inc
4 16 48 29 61
nth_prime5_inc
5 29 91 39 101
Alu 5 5 17 10 22
4_49 4 16 52 22 58
Hwb4 4 12 28 15 31
Hwb5 5 24 96 38 110
Hwb6 6 32 128 40 134
Pprm1 4 9 33 14 38
47
48
• Fig. 2. Circuit for function FX2 created with our method for traditional cost function calculation that does not take Ion Trap technology constraints into account.
49
abcd
0
1
1
1
0
1
a1
ad1
bab1
d
a aabdb
adabddb1
bddab
bcdcdacbcabdaddb1
garbage
garbage
garbage
garbage
garbage
function
Nearest Linear Node ModelNearest Linear Node Model
• Fig. 3. Circuit from Figure 2 modified with adding SWAP gates for new cost function calculation that does take Ion Trap technology constraints into account, with XX gates added. It has 36 SWAP gates added to realize LNNM. 50
All gates are realized only on neighbors, but we have to add many SWAP gates
51
bcdcdbcacabdbdad 1
abdbdad 1 bddba
abdba abb 1
ad1 a1
d
a
1 c
1 d 1 d
1 b 1 b
1 a 1 a 1 a
1 d
1
11
1
1 1
10
0
Example of Positive Davio Lattice from [Perkowski97d]. Positive Davio Expansion is applied in each node. Variable d is repeated
52
1
2 3
45 6
7 8 9
a
b
c
f
Garbage
Garbage
Garbage
a
b
c
f
Garbage
Garbage
Garbage
(a)
(b)
Transformation of function F3(a,b,c) from classical Positive Davio Lattice to a Quantum Array with Toffoli and SWAP gates. Each SWAP gate is next replaced with 3 Feynman gates.(a) intermediate form, (b) final Quantum Array.
Intermediate Structure with Dipal Gate
1
2 3
45 6
7 8 9
a
b
c
f
Garbage
Garbage
Garbage
Garbage
Garbage
Garbage
53
Another Representation of Quantum Array with Dipal Gate
a
b
c
f
Garbage
Garbage
Garbage
Garbage
Garbage
Garbage
d1
d2
d3
d4
d5
d6
x
y
z
v
54
Layered Diagram using Dipal Gate
a
b
55
General layout of the layered diagram
Each box represents a gate from family of Dipal gate
General Pattern of Circuit with Dipal Gate
d3
d6
d2
d4
d1
a
b
cx
y
z
v
d5
56
Quantum cost based On 1d model
57
Benchmark #Gates Lattice
Cost Lattice
#Gates with SWAP insertion for Lattice
Cost with SWAP gates for Lattice
#Gates DMM
Cost DMM
#Gates with SWAP insertion for MMD
Cost with SWAP gates for MMD
2to5 31 107 61 197 15 107 31 155
rd32 4 8 8 20 4 8 6 14
rd53 11 39 44 138 16 75 72 273
3_17 10 21 14 33 6 12 8 18
6sym 34 150 56 216 20 62 78 236
5mod5 14 58 17 67 10 90 48 204
4mod5 6 18 10 30 5 13 11 31
Ham3 3 7 3 7 5 7 7 13
Xor5 4 4 4 4 4 4 4 4
Xnor5 5 5 5 5 -------- -------- -------- --------
decod24 10 30 14 42 -------- -------- -------- --------
Cycle10_2 180 860 306 1238 19 1198 199 1738
Ham7 22 58 30 112 23 81 79 249
Quantum cost based On 1d model
58
Benchmark #Gates Lattice
Cost Lattice
#Gates with SWAP insertion for Lattice
Cost with SWAP gates for Lattice
#Gates DMM
Cost DMM
#Gates with SWAP insertion for MMD
Cost with SWAP gates for MMD
Graycode6 5 5 5 5 5 5 5 5
Graycode10 9 9 9 9 9 9 9 9
Graycode20 19 19 19 19 19 19 19 19
Nth_prime3_inc
4 6 5 9 4 6 6 12
Nth_prime4_inc
16 48 20 60 12 58 18 76
Nth_prime5_inc
29 91 39 121 26 78 128 384
Alu 5 17 7 23 -------- -------- ---------- ----------
4_49 16 52 41 127 16 58 40 130
hwb4 12 28 15 40 17 63 39 129
hwb5 24 96 44 156 24 104 64 224
hwb6 32 128 72 248 42 140 144 446
pprm1 9 33 19 63 -------- -------- ---------- ----------
GENERALIZED GENERALIZED REGULARITIES FOR REGULARITIES FOR
QUANTUM AND NANO-QUANTUM AND NANO-TECHNOLOGIESTECHNOLOGIES
59
Ion-Trap Layout
60
(a) (b) (c)
(d)
Single ion
Interaction between two ions
Various regular structures are technically possible, single dimensional vector is the one that is most often discussed
61
Examples of Expansions for regular Examples of Expansions for regular structuresstructures
Non-symmetric functions require Non-symmetric functions require repeatition of input variablesrepeatition of input variables
62
• Variable b is repeated
63
Symmetry Symmetry Indices and Indices and
regular regular structures for structures for binary logicbinary logic
64
EXAMPLE: EXAMPLE:
MULTI-VALUED MULTI-VALUED REVERSIBLE LOGIC REVERSIBLE LOGIC
ADDERADDER
65
MULTI-VALUED REVERSIBLE LOGICMULTI-VALUED REVERSIBLE LOGIC
66
67
68
Three dimensional Three dimensional realization of realization of lattices for ternary lattices for ternary logic: SUMlogic: SUM
69
Three dimensional Three dimensional realization of realization of lattices for ternary lattices for ternary logic: CARRYlogic: CARRY
70
QUANTUM CIRCUITS QUANTUM CIRCUITS AND QUANTUM AND QUANTUM
ARRAYS FROM TRULY ARRAYS FROM TRULY QUANTUM GATESQUANTUM GATES
71
Binary Reversible Gates
Basic single qubit quantum gates
72
NOT Pauli x Pauli y Pauli z Hadamard
X Y Z H
01
10x
0
0
i
iy
10
01z
11
11
2
1H
XX YY ZZ HH
(a) (b) (c) (d)
01
10x
0
0
i
iy
10
01z
11
11
2
1H
Phase gate Pseudohadamard gate Inverse pseudohadamard gate
V S h 1h
ii
iiiV
11
1
2
1
ie
S0
01)(
11
11
2
1h
11
11
2
11h
V Gate
VV SS hh 1h 1h
ii
iiiV
11
1
2
1
ie
S0
01)(
11
11
2
1h
11
11
2
11h
(a) (b) (c) (d)
73
• The The transformations transformations
of blocks of of blocks of quantum quantum gates to the gates to the pulses level.pulses level.
• Transformation of the circuit realized in Fig. 7 using Toffoli gate. Each Toffoli and SWAP gates are replaced by quantum CNOT and CV/CV+ quantum gates and rearranged to satisfy the neighborhood requirements of Ion trap.
•
74
Lattice based Lattice based FPGA in FPGA in
CLASSICAL CLASSICAL LOGICLOGIC
75
New type of FPGA in CMOSNew type of FPGA in CMOS
In classical CMOS logic one can design a regular array, such as a form of FPGA, which realizes Shannon, positive Davio and negative Davio inside one cell.
Such array is highly testable
We can try to design something similar in quantum and reversible logic circuits.
76
Design of SRFPGA cellDesign of SRFPGA cell
77
1. Dipal completed his MS in December 2000 with thesis on “Method for Self-Repair of FPGAs”.
2. I adapted concept of Lattices which were developed Dr. Perkowski and Dr. Jeske to design FPGA like regular structure in VLSI
This cell can be mapped to Shannon, positive Davio, negative Davio and other logic gates.
General idea of SRFPGA architecture
78
•General idea of the SRFPGA architecture, each circle represents cell shown on the previous foil.•Row and column decoders are for memory addressing•The next foil shows actual physical design of the SRFPGA
79
SRFPGA layoutWith I/O pins
80
Var1
var2
var3
var4
var5
var6
var7
var8
var9
var10
var11
var12
var13
var14
var15
var16
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
0
Input test vector
Inputtestvector
Test output
Testoutput
Faults observed during column testC = 2.
Faults observed during
diagonal testD = 2
Total number ofFaults N = C * D= 2 * 2 = 4.
1. Dipal developed a unique test that identifies any number of faulty cell in the FPGA
2. Repair is based on redundancy-repair where identified faulty cells are replaced with unused good cell in the structure
3. Later Dipal adapted concept of lattice and synthesis methodology for designing reversible logic circuits.
4. His method of reversible circuit design resolves many issues that are not yet addressed by any other researchers
5. This approach can be extended to reversible and quantum logic cicuits.
CONCLUSIONS and CONCLUSIONS and possible projectspossible projects
82
Conclusions Experimental results proved that our algorithm produced better
results in terms of quantum cost compared to other contemporary algorithms for synthesis of reversible logic.
New gate family called Dipal gate
Presented new synthesis method with layered diagrams. More accurate technology specific cost model for 1D qubit
neighborhood architecture.
83
CONCLUSIONS A new method based of lattice diagram to synthesize reversible logic circuit with 3x3 Toffoli gates.A new method based of lattice diagram to synthesize reversible logic circuit with 3x3 Toffoli gates.
A new family of gates called Dipal Gates.A new family of gates called Dipal Gates.
New diagrams called layered diagram that uses family of Dipal gate for synthesis of reversible logic New diagrams called layered diagram that uses family of Dipal gate for synthesis of reversible logic
function.function.
Software for creating Lattice diagrams and software for creating quantum array from Lattice (Lattice Software for creating Lattice diagrams and software for creating quantum array from Lattice (Lattice
to QA).to QA).
Program to implement a variant of MMD algorithm.Program to implement a variant of MMD algorithm.
84
Possible ProjectsPossible Projects
1.1. Generalize to ternary logicGeneralize to ternary logic
2.2. Generalize to all Dipal Gate Family gates.Generalize to all Dipal Gate Family gates.
3.3. Realization with low level pulses for NMR technology.Realization with low level pulses for NMR technology.
4.4. Development of a concept of reversible/quantum FPGA Development of a concept of reversible/quantum FPGA similar to SRFPGAsimilar to SRFPGA
5.5. Extend Agrawal-Jha method for factorized circuits.Extend Agrawal-Jha method for factorized circuits.
6.6. Extend the methods to many-output circuits.Extend the methods to many-output circuits.
85
What to remember?What to remember?1. Use of PPRM in synthesis of reversible circuits.2. The main idea of Agrawal-Jha algorithm.3. How AJ algorithm can be improved?4. How this algorithm can be extended to Fredkin gates?5. Expansions Rules for Lattice Diagrams6. Creating Positive Davio Lattice7. Creating Negative Davio Lattice8. Creating Lattice for arbitrary function with a mixture of
Davio and Shannon Expansions.9. Lattices for symmetric functions.10. Transforming Positive Davio Lattice to a quantum array
(circuit) for single output functions. 86
What to remember?What to remember?1. Transforming Positive Davio Lattice to a quantum array
(circuit) for single output functions.2. Transforming Positive Davio Lattice to a quantum array
(circuit) for multi-output functions.3. Dipal gate and Dipal gate family.4. Regular structures and their use in quantum computing.5. Regularity versus LNNM model.6. Multiple-valued Lattices for ternary logic.7. FPGA based on 3*3 lattices and can they be adapted to
quantum and reversible circuits.8. Decomposition to pulses. Relation to quantum costs.
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